Quasi-Perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension

A construction of two-quasi-perfect Lee codes is given over the space ?np for p prime, p ? ±5 (mod 12), and n = 2[p/4]. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee metric do not exist for dimension n ? 3 and radius r ? 2. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between coding and graph theories

50 Years of the Golomb--Welch Conjecture

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.Comment: 28 pages, 2 figure

Geometric approach to sampling and communication

Relationships that exist between the classical, Shannon-type, and geometric-based approaches to sampling are investigated. Some aspects of coding and communication through a Gaussian channel are considered. In particular, a constructive method to determine the quantizing dimension in Zador's theorem is provided. A geometric version of Shannon's Second Theorem is introduced. Applications to Pulse Code Modulation and Vector Quantization of Images are addressed.Comment: 19 pages, submitted for publicatio

Filter and nested-lattice code design for fading MIMO channels with side-information

Linear-assignment Gel'fand-Pinsker coding (LA-GPC) is a coding technique for channels with interference known only at the transmitter, where the known interference is treated as side-information (SI). As a special case of LA-GPC, dirty paper coding has been shown to be able to achieve the optimal interference-free rate for interference channels with perfect channel state information at the transmitter (CSIT). In the cases where only the channel distribution information at the transmitter (CDIT) is available, LA-GPC also has good (sometimes optimal) performance in a variety of fast and slow fading SI channels. In this paper, we design the filters in nested-lattice based coding to make it achieve the same rate performance as LA-GPC in multiple-input multiple-output (MIMO) channels. Compared with the random Gaussian codebooks used in previous works, our resultant coding schemes have an algebraic structure and can be implemented in practical systems. A simulation in a slow-fading channel is also provided, and near interference-free error performance is obtained. The proposed coding schemes can serve as the fundamental building blocks to achieve the promised rate performance of MIMO Gaussian broadcast channels with CDIT or perfect CSITComment: submitted to IEEE Transactions on Communications, Feb, 200

On the Peak-to-Mean Envelope Power Ratio of Phase-Shifted Binary Codes

The peak-to-mean envelope power ratio (PMEPR) of a code employed in orthogonal frequency-division multiplexing (OFDM) systems can be reduced by permuting its coordinates and by rotating each coordinate by a fixed phase shift. Motivated by some previous designs of phase shifts using suboptimal methods, the following question is considered in this paper. For a given binary code, how much PMEPR reduction can be achieved when the phase shifts are taken from a 2^h-ary phase-shift keying (2^h-PSK) constellation? A lower bound on the achievable PMEPR is established, which is related to the covering radius of the binary code. Generally speaking, the achievable region of the PMEPR shrinks as the covering radius of the binary code decreases. The bound is then applied to some well understood codes, including nonredundant BPSK signaling, BCH codes and their duals, Reed-Muller codes, and convolutional codes. It is demonstrated that most (presumably not optimal) phase-shift designs from the literature attain or approach our bound.Comment: minor revisions, accepted for IEEE Trans. Commun